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Prime Suspect

That’s the title of episode five, season one. It starts out looking like a kidnapping, but it turns into a high stakes caper involving some totally ruthless people and some classic math. I caught this on Amazon Prime Video. Details are from Wikipedia and IMDB. It’s NUMB3RS, and it’s ultimately about math.

Ethan (Neil Patrick Harris) and Becky (Susan Egan) Burdick are hosting a birthday party for their daughter Emily (Emma Prescott). As the party is winding down things turn sour when the clown tosses Emily into his van and drives off. The FBI will get involved.

Don Eppes (Rob Morrow) is the agent in charge of the case. His brother Charlie (David Krumholtz), is a math prof at a local college, who comes in to help the FBI where math is involved.

I’m not going to detail the plot. However, there is some Skeptical Analysis involved.

It’s a kidnapping for ransom, but for an unusual ransom. Ethan Burdick is a math genius who is working on a resolution of the Riemann Hypothesis, named after Bernhard Riemann who first posed it in 1859. Ethan is supposed to have a solution to the (then) 150-year-old problem. It’s the solution the crooks are after. They will trade the child for the answer.

Despite the clown makeup, Agent Terry Lake (Sabrina Lloyd) recognizes the clown as a criminal recently sprung from stir. Bad news, the family has been warned not to deal with the police, so the FBI has to go it alone.

Except that Charlie meets with Ethan, and the two of them go over Ethan’s resolution of Riemann. Charlie’s observation is heart-wrenching. Ethan is on the wrong track. He has no resolution for the Riemann Hypothesis. He has nothing to give the crooks.

Charlie and the agents also discover what the crooks are after. They will use the resolution of the Riemann Hypothesis to crack the encryption key used by the Federal Reserve. They will obtain the Prime Interest Rate hours in advance of its release and will bet heavily on related investments. And win.

What Charlie, Ethan, and the FBI do is to work with the Federal Reserve and concoct a fake key leading to a fake page. Ethan will give the crooks the method, the crooks will compute the key and use the key to access the fake page. The feds will be watching all this activity and will pinpoint the location of the cracking operation. They plan to move in before the crooks can scram.

Not known, but suspected, is that once the crooks have the Prime Rate, they won’t need Emily any longer nor the math prof working with them.

The trap is sprung, the feds move in like gangbusters, one of the miscreant’s takes a load to the chest from an agent, and Emily is snatched from the arms of the ring leader in exchange for his life.

Except that this is not exactly what the Riemann Hypothesis is about. Follow the link above and read all about it. In short, Riemann proposed to be able to identify limits to the population of prime numbers within a bounded domain. What does that have to do with cracking the key to the Federal Reserve server? Glad you asked:

An essential requirement of a public key system is that your everyday Edward Snowden should not be able to take E and derive D from it. The method described by RSA involves using pairs of very large prime numbers. Call a pair of these numbers p and q. Then

p x q = n.

The number n is not prime. It has only two factors, p and q. Now suppose each of p and q are 100 decimal digits long (or more). Then the length of n is 200 (or more). The RSA method uses p and q (and n) to produce e and d. Read the RSA paper, page 6. This involves some nice math, which I will not elaborate on here.

A user R can publish n and e, keeping d (and p and q) private. Somebody wanting to send R a message uses n and e to encrypt the message. R uses n and d to decrypt the message. Knowing n it is still very difficult to compute d, even if you know e. Computing d is tantamount to factoring n (into p and q). It is well known that the factoring problem is hard. Factoring n is only a bit less difficult than doing a search for p (or q), but it is not easy enough to make it feasible with present day computational facilities.

And there is more interesting stuff. You can go to my previous post and read up on it. The essential point is this. For a public key system, the person owning the key makes public his key for encryption. However, the decryption key, which the owner holds private, is required for reading the encrypted message. People use his encryption key to code messages, and they send messages to the key owner. He is the only one with the decryption key, and he is the only one who can read messages encrypted with his key.

The deal is this. The public key is a large integer, maybe hundreds of digits long. It’s a composite, the product of two prime numbers. If you can factor the encryption key and obtain the two prime factors, then you can compute the decryption key and read all the secret mail sent to the owner.

This is not to say the Fed uses this method, but if they did, they might do the following:

Determine the new Prime Rate, a few days in advance. And keep it a secret.

Pput the Prime Rate information on a secure server. A secret key would be required to access the server and read the new Prime Rate.

The public key system is not used for sending lengthy messages. It’s only practical for sending short strings, such as the key to the Fed server.

The gatekeeper of the Fed server would use the public keys of various parties (hopefully only Fed employees) to send them the pass code for the Prime Rate page.

Now, the crooks count on having one of the public keys in question. These keys really are public. If they can factor one of these public keys they can possibly eavesdrop on a communication link and obtain the encrypted pass code. The crooks compute the associated private key and read the pass code. Then they use the pass code to read the (unpublished) Prime Rate and get ready to make a lot of money in just a few hours.

Anyhow, that’s how it might work. Except that Riemann’s Hypothesis does not seem to be a key to factoring large prime numbers. For one thing, it’s a hypothesis. They conclusion is stated in the hypothesis. Everybody already knows what the conclusion is. The big deal about the Riemann Hypothesis is to prove that it’s true.

As a side note, the Riemann zeta function was an object of Alan Turing’s interest in constructing a computing machine. A good book about this episode is by Andrew Hodges, Alan Turing: The Enigma. The link is to my review of the book. A first rate movie, The Imitation Game, is based on the book, and a review is scheduled to be out next month.